Application of Bayesian Random Coefficient Models in Modeling traffic Congestion

ABSTRACT

• Analytical models developed using field traffic data can be utilized with acceptable confidence to reliably represent, predict, and evaluate the operational characteristics of the highway systems.
• Establishing a model to estimate the stochastic evolution of traffic conditions is important for prediction and developing effective solution to address roadway congestion.
• Using a freeway segment of I-295 located in Jacksonville, Florida as a case study, an advanced statistical method developed in the Bayesian modeling framework was developed to account for the day of the week and lane lateral location disparities effect in influencing the dynamic evolution characteristics of traffic regimes.
• Specifically, this study applied the Markov Chain theory and random coefficients logistic regression model in the analysis.

Introduction

• Random coefficient models best fit data which are nested/clustered

• The correlation within the group can be analyzed

• Unlike indepencence assumption in basic regression models

• Also disperities between the groups can be quantified

Random coefficient models

• AKA:

  • Hierarchical models

  • Multilevel models

  • Mixed-effect models

  • Random-effect models

  • Variance-component models

  • Partial pulled models

Motivating Examples

• Traffic data like many other social science data are nested in nature
  • Vehicles in different lanes
  • Traffic in different days of the week, etc
• As a results disparties arise in data characteristics
  • May be Monday has different pattern than Wendsday
  • May be inside lanes (near median) has few vehicles than outside lane (near shoulder)
• Models without accounting these effect lead to inconsistency conclusion

Random coefficient vs. Basic model

In the absence of a Bayesian hierarchical model, there are two approaches for this problem:

• Independently compute influence of factors on the responce variable for each group (no pooling)

• Compute an overall average, under the assumption that every group has the same underlying average (complete pooling)

Complete pooling and No pooling

• Data observations are independent

• Underlying characteristics is combined to a grand mean

• One homogeneous model is fitted

• Random coefficient provide a tradeoff between no pooling and complete pooling

• Resulting model also known as Partial pooled model

Partial pooled model/Random coefficient

• Data observations are nested to their group

• Underlying characteristics are nested in groups

• Heterogeneous models are fitted

Research Objectives

• Modeling the dynamic characteristics of traffic conditions

• Investigate disperity associated with lane lateral loaction

• Also, the effect of day of the week

Methodology

Dynamic Modeling of traffic conditions

• Markov Chain theory was used in the analysis

  • Only current traffic regime has influence on the next regime

  • Mathematically:

\[P_{ij}=(P_{t+1}=j|P_{t}=i)\]

\[P_{ij} = \left[\begin{array} {rr} P_{ff} & P_{fc} \\ P_{cf} & P_{cc} \\ \end{array}\right] \]

• Dynamic Characteristics

  

• Markovian Property

\[\sum_{j=1}^{2} P_{ij}=1\]

\[P_{ij} = \left[\begin{array} {rr} 1-P_{fc} & P_{fc} \\ 1-P_{cc} & P_{cc} \\ \end{array}\right] \]

Issues Addressed

Time-varying effect

• Regression to model Markov chain Transition probability

  • Model is more flexible

  • Heterogeneous issue by time is addressed

Lane and day of the week varying-effect

• Random coefficient models

  • Address heterogeneous characteristics associated by lane lateral location

  • Address heterogeneous characteristics associated by day of the week

Modeling Approach

• Four series of random coefficient models were developed:
  • Varying Intercept Model
  • Varying Intercept and Constant Slope Model
  • Constant Intercept and Varying Slope Model
  • Varying Intercept Model and Varying Slope Model

Model 1

• Varying Intercept Model and Varying Slope Model

\[ Y_{ij} = Bernoulli(\pi_{ij})\] \[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \beta X + \epsilon_{k}\]

• Prior

\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]

• Prior

\[ \sigma_{1} \sim unif(0, 100)\] \[ \beta \sim N(\mu=0, \sigma = 100)\] \[ \epsilon_{k} \sim N(\mu=0, \sigma = \sigma_{k})\] \[ \sigma_{k} \sim halfcauchy(0, 5)\]

Model 2

• Varying Intercept Model

\[ Y_{ij} = Bernoulli(\pi_{ij})\]
\[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \epsilon_{k}\]

• Prior

\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]

• Prior \[ \sigma_{1} \sim unif(0, 100)\] \[ \epsilon_{k} \sim N(\mu=0, \sigma = \sigma_{k})\] \[ \sigma_{k} \sim halfcauchy(0, 5)\]

Model Comparison

• Widely Information criterion (WAIC)
• Penalize model with large number of parameters
• Fully Baseyian approach - uncertanity incorporated

\[ WAIC = -2*lppd + 2*pWAIC\]
where,

          lppd is the log point-wise posterior density
          pWAIC is the effective number of parameters

Results

• Estimated Speed threshold

Random coefficient	Lane lateral location
Median lane	46.72
Inner-left lane	51.42
Inner-right lane	50.93
Shoulder lane	57.26
Overall	52.23

Results

• Speed occupancy relationship

  

Results

• Speed occupancy relationship

  

Results

• Speed occupancy relationship

  

• Speed occupancy relationship

  

Results

• Model Comparison Results

• No significant difference in model fit between CIRS and RC for the SCR data

• RIS was the best in fitting Breakdown data

RC was selected for modeling Breakdown and SCR data

Results

• Model Comparison Results

Results

• Remaining in the congested regime
  

• Comparison of the estimated probability at different flow rate

  

  Results

• Breakdown probability

  

• Comparison of the estimated probability at different flow rate

  

  Results

• Symmetrical properties of logistic model

  

• Recovery process

  

• Comparison of the estimated probability at different flow rate

  

  Conclusions

  • The results show that there are considerable evidence that there are heterogeneity characteristics of evolution characteristics associated with different lane lateral location.

• However, the day of the week influence on the disparities of the estimated transition phenomenon was found only to influence the breakdown transition process.
• The lane lateral location disparity effect was quantified and found that shoulder lane is prone to both breakdown or SCR compared to lanes near the median.
• The results of the random coefficient logistic model in estimating the Markov Chain transition probability show that flow rate is significant at 95% credible interval (CI) in predicting the likelihood of transition from one traffic regime to the next.
• For breakdown process, an “S” shaped pattern was revealed for the relationship between the flow rate and the breakdown probability while an increasing “concave downward” shaped pattern was found in the process of staying in the congested regime (SCR).